Edit in JSFiddle

var i=1,
    container=$('<div>'),
    addToList=$('#addToList');

addToList.after(container);
container.append('<select id="dropDown" size="10" style="width:200px">');

addToList.click(function(){
    $("#dropDown").append($('<option />').text('Item '+i++));
    fixDropDown(document.getElementById("dropDown"));
});   

var fixDropDown=function(control){
    var option = document.createElement("option");
    control.add(option, null);
    control.remove(control.options.length - 1)  
};

<input type="button" id="addToList" value="Add to List" />